Solid cores of neutron stars 771 Physical origins and models

The history of theories of hypothetical crystalline neutron star cores, sketched in § 7.2.4, started with the idea of crystallization of neutron matter owing to the strong short-range repulsion of the neutron-neutron interaction. This repulsion was often represented by a hard-core one, so that vij = x> for Tij < Tcore. The hard-core repulsion solidifies the neutron matter at nb ~ (4^T3ore/3)-1; in this case neutrons become localized by the infinite potential walls. Other arguments in favor of crystallization were taken from the physics of noble gases. The short-range repulsion between two rare-gas atoms is well represented by the repulsive term of the Lennard-Jones potential vij k (rij)-12. It is well known that noble gases solidify at sufficiently high pressures. In the beginning of the 1970s some authors applied the so-called law of corresponding states, first suggested by Anderson & Palmer (1971), to deduce the solidification density of neutron matter by scaling the experimental results for 3He.

However, the short-range neutron-neutron repulsion occurs due to the exchange of vector mesons (see § 5.6). Therefore, it is of the Yukawa form vij k exp(—irij)/ (irij), often called a "soft-core repulsion". In this case the hard-core and noble-gas arguments are invalid. Moreover, many-body calculation of the liquid-solid transition in dense neutron matter is a tremendous computational challenge. One has to calculate E(liq) and E(sol) with very high precision in order to find the density above which E(liq) > E(sol). By 1974, the precision was still insufficient. Some calculations gave no solidification at all, while several authors obtained solidification at (5 — 30) x 1014 g cm-3 (the review of the early, alas, unreliable calculations is given by Canuto 1975). The consensus was reached in the second-half of the 1970s, when the precision of many-body calculations became sufficiently high: for realistic vij neutron matter does not solidify at densities expected in neutron star cores.

Another possibility of getting a solid structure is to increase the energy from the medium-range attractive tensor component (§5.5.1). This mechanism, related to the n0-condensation in neutron matter (§ 7.3), was analyzed by Pandharipande & Smith (1975). The amplification of an already strong tensor interaction component was induced by the coupling of nucleons and pions to A-isobars excited in intermediate nucleon states. Strong coupling at the nAn vertex amplified the tensor component of the many-body Hamiltonian, VT. Its attractive contribution became sufficient to arrange neutrons in a cubic lattice and correlate their spins with crystal planes. The spin-quantization axis was chosen to be parallel to the cube side. All neutrons in a given plane perpendicular to the quantization axis had the same spin projection. Neutrons in adjacent planes had antiparallel spins. Such a spin and space structure gave a large attractive contribution to E from the tensor interaction component

i<j where nij = rij/rij (see §5.5.1). The value of E(sol) was then obtained variationally by minimizing the expectation value of the Hamiltonian within a family of neutron wave functions, localized around the crystal lattice sites. Those wave functions, with alternating spin polarization in adjacent crystal planes, took full advantage of the tensor attraction (in contrast to the disordered liquid state, in which the first-order tensor contributions average to zero). In this way Pandharipande and Smith obtained E(sol) < E(liq) for p > 3p0. The first-order liquid-solid phase transition was associated with a density jump, by ~20%, which considerably softened the EOS. Because of its periodic spatial and spin structure, the ground state wave function led to a nonzero expectation value of the n0-condensate. The neutral pion condensate had a standing-wave structure with a characteristic wave number kno (see § 7.3).

The first-order contribution VT?PE of the one-pion exchange potential (OPEP, see § 5.6) vanishes in an isotropic fluid. However, as shown by Takat-suka & Tamagaki (1976, 1977), it can dominate in a state of the so called alternating-spin layers (ALS, see below; also see Takatsuka et al., 1978; Mat-sui et al., 1979). In the ALS state, neutrons and protons are localized in parallel planes (one dimensional, 1D, localization). Let the planes be perpendicular to the spin quantization z-axis. Neutrons localized in one plane have spins aligned with the z-axis and protons in this plane have opposite spins, so that the isospinspin states are (n t) and (p In adjacent planes one has (n and (p t). As we are dealing with the 1D localization (nucleons can freely move in the xy plane), matter behaves as a smectics A phase in the nomenclature of liquid crystals (see § 3.7.2). If the binding gain due the tensor attraction overcomes the binding loss due to the kinetic energy increase (resulting from 1D localization), the dense matter undergoes the phase transition to the ALS-phase.8 Calculations of the ALS structure of dense matter were further developed by the Kyoto

8 The idea that a strong tensor component of the OPEP can lead to a solid-like structure of nucleon matter with a periodic spin-isospin ordering was first proposed by Calogero et al. (1973); also see Calogero et al. (1975).

group, with the emphasis on possible signatures of this phase in neutron stars (see Takatsuka et al. 1993 for review).

Moreover, Kutschera & Wojcik (1989, 1990) suggested that, for a sufficiently low proton fraction xp < 0.05, nuclear matter could be unstable with respect to proton localization accompanied by a modulation of neutron density. At such low xp, protons behave as impurities in neutron matter interacting mainly with the neutron background. While localization increases the proton kinetic energy, it increases also the proton binding in neutron matter by locking protons in potential wells corresponding to minima of the neutron density. Further work pointed out an analogy with polarons, well known in the physics of condensed matter.9 Protons seem to behave as nuclear polarons and form a lattice at high densities (Kutschera & Wojcik, 1993, 1995). Of course, the proton localization occurs only for those model EOSs which predict the decrease of the proton fraction xp with growing p at high densities. Moreover, calculations of E(sol) involve many approximations. Proton localization takes place if the difference E(sol) —E(liq) becomes negative for p > ploc. Calculated difference turns out to be so small that numerical results should be taken with a grain of salt. Recent calculations, performed for several models of the npe^ matter [with decreasing xp at high p and vanishing protons at still higher p!] give ploc = (3 — 6) po (Kutschera et al., 2002). The idea of localization of impurities was extended to the hyperonic matter by Perez Garcia et al. (2002).

In the mixed-phase state of dense matter (Pa™' < P < Pgm), see § 7.6) each of the phases is electrically charged. To minimize the sum of the Coulomb and surface energies, the less dense and more dense phases (A and B) can be distributed into a periodic structure (see Glendenning 2001 and references therein). For a low volume fraction % ^ 1 of phase B, spherical droplets of this phase can form a cubic [most probably, body-centered cubic (bcc)] lattice immersed in the background of phase A. With increasing %, the three dimensional (3D) cubic lattice is thought to be replaced by the two dimensional columnar phase of B-rods immersed in phase A, analogous to the 2N phase in the bottom of the neutron star crust (§ 3.4.2). The difference from the crust is that the hadron component of phase A is electrically charged. For % ~ 1/2,1D alternating A and B slabs appear; at higher % they are replaced by the columnar phase of A-rods immersed in the liquid of phase B. At % ^ 1, droplets of phase A form a bcc crystal in such a liquid.

9The formation of polarons in terrestrial crystals results from the electron-phonon interaction, which can lead to a strong increase of the electron effective mass. The strongest effect occurs in ionic crystals because of the strong Coulomb attraction between ions and electrons. Electrons can become self-trapped in local deformations of the ion lattice (see, e.g., Chapter 10 of Kittel 1986). The instability of crystals against density perturbations in terrestrial solids corresponds to the instability of the neutron component in the npe^ matter.

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